Abstract
(Global) optimization is one of the fundamental challenges in scientific computing. Frequently, one encounters objective functions or search space topologies that do not fulfill necessary requirements for well understood and efficient procedures like, e.g., linear programming. This methodological gap is filled by metaheuristic optimization approaches. Their search dynamics in high dimensional search spaces and for complicated objective functions is not well understood at present. In particular, the choice of parameters driving the procedures is a demanding task. In this contribution we show how insight from time series analysis help to investigate – on a pure empirical basis – metaheuristic schemes. Rather than deriving analytical results on convergence behavior, ex ante, we propose online observation of the search and optimization progress. To this end, we use the Detrended Fluctuation Analysis – a method from time series analysis – to investigate the search dynamics of metaheuristics as stochastic processes. We apply the proposed method to two different metaheuristic, namely differential evolution and basin hopping.
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References
Bäck, T., Schwefel, H.: An overview of evolutionary algorithms for parameter optimization. Evolutionary Computation 1(1), 1–23 (1993), http://dx.doi.org/10.1162/evco.1993.1.1.1
Binder, K., Heermann, D.: Monte Carlo Simulation in Statistical Physics, 3rd edn. Springer, Berlin (1997)
Binder, K., Young, A.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58(4), 801–976 (1986)
Birattari, M.: Tuning Metaheuristics. SCI, vol. 197. Springer, Heidelberg (2009)
Bunde, A., Kantelhardt, J.: Langzeitkorrelationen in der natur: von klima, erbgut und herzrhythmus. Phys. Bl. 57(5), 49–54 (2001)
Chou, C., Hand, R., Li, S., Lee, T.: Guided simulated annealing method for optimization problems. Phys. Rev. E 67, 66704 (2003)
Das, S., Suganthan, P.: Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation 15(1), 4–31 (2011)
Doye, J., Wales, D.: Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids. J. Chem. Phys. 116(9), 3777–3788 (2002)
Friedrich, T., Kroeger, T., Neumann, F.: Weighted preferences in evolutionary multi-objective optimization. In: Wang, D., Reynolds, M. (eds.) AI 2011. LNCS, vol. 7106, pp. 291–300. Springer, Heidelberg (2011)
Friedrich, T., Sauerwald, T.: The cover time of deterministic random walks. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 130–139. Springer, Heidelberg (2010)
Hamacher, K.: On stochastic global optimization of one-dimensional functions. Physica A 354, 547–557 (2005)
Hamacher, K.: Adaptation in stochastic tunneling global optimization of complex potential energy landscapes. Europhys. Lett. 74(6), 944–950 (2006)
Hamacher, K.: Adaptive extremal optimization by detrended fluctuation analysis. J. Comp. Phys. 227(2), 1500–1509 (2007)
Hamacher, K., Wenzel, W.: The scaling behaviour of stochastic minimization algorithms in a perfect funnel landscape. Phys. Rev. E 59(1), 938–941 (1999)
Hansmann, U., Wille, L.T.: Global Optimization by Energy Landscape Paving. Phys. Rev. Lett. 88(23), 68105 (2002)
Hoos, H., Stützle, T.: On the empirical evaluation of Las Vegas algorithms (1998)
Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Eugene Stanley, H.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64(1), 011114 (2001)
Jack, W., Rogers, J., Donnelly, R.A.: Potential transformation methods for large-scale global optimization. SIAM Journal on Optimization 5(4), 871–891 (1995), http://link.aip.org/link/?SJE/5/871/1
Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220, 671–680 (1983)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)
Panos, M., Pardalos, D.S., Xue, G. (eds.): Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding. dIMACS workshop, March 20-21. DIMACS – Series in Discrete Mathematics and Theoretical Computer Science, vol. 23 (1995)
Pardalos, P.M., Shalloway, D., Xue, G.: Optimization methods for computing global minima of nonvoncex potential energy functions. J. Glob. Opt. 4, 117–133 (1994)
Pardalos, P., Romeijn, E., Tuy, H.: Recent developments and trends in global optimization. J. Comp. Appl. Math. 124(1-2), 209–228 (2000)
Pellegrini, P., Stützle, T., Birattari, M.: Off-line vs. on-line tuning: A study on MAX-MIN ant system for the TSP, pp. 239–250 (2010)
Peng, C.K., Buldyrev, S., Havlin, S., Simons, M., Stanley, H., Goldberger, A.: Mosaic organization of dna nucleotides. Phys. Rev. E 49, 1685 (1994)
Ratschek, H., Rokne, J.G.: Efficiency of a global optimization algorithm. SIAM Journal on Numerical Analysis 24(5), 1191–1201 (1987), http://link.aip.org/link/?SNA/24/1191/1
Schelstraete, S., Schepens, W., Verschelde, H.: Energy minimization by smoothing techniques: a survey. In: Balbuena, P., Seminario, J. (eds.) Molecular Dynamics: From Classical to Quantum Methods, Amsterdam, pp. 129–185 (1999)
Schöbel, A., Scholz, D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Global Optimization 48(3), 473–495 (2010)
Shi, Y.-j., Teng, H.-f., Li, Z.-q.: Cooperative co-evolutionary differential evolution for function optimization. In: Wang, L., Chen, K., S. Ong, Y. (eds.) ICNC 2005. LNCS, vol. 3611, pp. 1080–1088. Springer, Heidelberg (2005)
Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9(3), 379–388 (1972)
Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G.: Exact ground states of ising spin glasses: New experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80, 487 (1995)
Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conference of the North American Fuzzy Information Processing Society (1996)
Storn, R., Price, K.: Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Opt. 11(4), 341–359 (1997)
Stützle, T.: Iterated local search for the quadratic assignment problem. European Journal of Operational Research 174(3), 1519–1539 (2006)
Sttzle, T., Hoos, H.H.: Analyzing the run-time behaviour of iterated local search for the TSP. In: III Metaheuristics International Conference. Kluwer Academic Publishers (1999)
Sutton, A.M., Neumann, F.: A parameterized runtime analysis of evolutionary algorithms for the euclidean traveling salesperson problem. In: Hoffmann, J., Selman, B. (eds.) AAAI, AAAI Press (2012)
Törn, A., Žilinskas, A.: Global Optimization. LNCS, vol. 350. Springer, Heidelberg (1989)
Wales, D.J., Scheraga, H.A.: Global Optimization of Clusters, Crystals, and Biomolecules. Science 285(5432), 1368–1372 (1999), http://www.sciencemag.org/cgi/content/abstract/285/5432/1368
Wenzel, W., Hamacher, K.: A Stochastic tunneling approach for global minimization. Phys. Rev. Lett. 82(15), 3003–3007 (1999)
Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1(1), 67–82 (1997)
Yang, X.S.: Metaheuristic optimization: Algorithm analysis and open problems. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 21–32. Springer, Heidelberg (2011)
Zemel, E.: Measuring the quality of approximate solutions to zero-one programming problems. Mathematics of Operations Research 6(3), 319–332 (1981)
Zlochin, M., Dorigo, M.: Model-based search for combinatorial optimization: A comparative study. In: Guervós, J.J.M., Adamidis, P.A., Beyer, H.-G., Fernández-Villacañas, J.-L., Schwefel, H.-P. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 651–661. Springer, Heidelberg (2002)
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Hamacher, K. (2014). Online Performance Measures for Metaheuristic Optimization. In: Blesa, M.J., Blum, C., Voß, S. (eds) Hybrid Metaheuristics. HM 2014. Lecture Notes in Computer Science, vol 8457. Springer, Cham. https://doi.org/10.1007/978-3-319-07644-7_13
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DOI: https://doi.org/10.1007/978-3-319-07644-7_13
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